Question

A cylinder of radius R has the y axis as its central axis, and therefore appears as a circle in the diagram below. The cylinder does not move. A massless string of total length b (greater than TR) is attached to the cylinder as shown in the diagram. A point mass m is attached to the lower end of the string. The motion is in the xz plane. The point P is defined so that the string above P is in contact with the cylinder, and the string elow P is not in contact with the cylinder, as shown in the diagram. This means the ocation of P on the string changes as m swings left and right. We will use the angle ? as our generalized coordinate for this problem (-t/2 < ??t/2). The acceleration due to the Earth's gravitational field is the familiar constant vector g pointing in the negative z direction. [Hint: It will be useful to define a new constant cb-TR/R (a) What is the length of the string not in contact with the cylinder, as a function of ?? (b) What are the x and z coordinates of the point mass m functions of ?? (c) Calculate the Lagrangian. (d) From the Lagrangian write out the equation of motion as a second order (e) Because the Lagrangian is time-independent, there is a conserved quantity. Write (f) The particle is released from rest at angle ? . Write an integral that gives the exact (g) Calculate the period of small oscillations about the equilibrium location. differential equation. You do not have to solve this equation. out this quantity. If you use a theorem you must state the theorem. value of the period in terms of this angle. You do not have to solve the integral Z string is attached here and this end p does not movw

Answer 1

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